British officials hope the emollient presence of the Queen in Berlin on the eve of a European summit will reinforce Ms Merkel’s willingness to help Mr Cameron keep the UK inside the EU.

“Queen Elizabeth is witness to the development of the relationship between Germany and Britain which is very friendly today,” Ms Merkel said this week ahead of the royal visit, which began on Tuesday night.

Friendly perhaps, but marred by a series of misunderstandings in which Mr Cameron has placed too much faith in Ms Merkel’s willingness or ability to fix his European problems.

On Thursday Mr Cameron will head to Brussels to set out his broad objectives for an EU renegotiation ahead of the UK’s planned referendum on membership, which Downing Street hopes to hold in 2016.

Mr Cameron hopes that Ms Merkel will back his plan to hand over detailed work on solving “the British question” to senior EU officials; he wants final proposals on the table at the December summit in Brussels.

No relationship in Europe is more important to the prime minister than the one with Ms Merkel — his officials refer to Germany as the G1 in the eurozone — but when put to the test it has often been found wanting.

Ms Merkel and Mr Cameron have shared interests in an open and competitive Europe and the two leaders have worked together in promoting trade deals, cutting the EU budget in 2013, and imposing sanctions on Russia over its incursion in Ukraine.

But on other issues things have not gone so well: Ms Merkel left Mr Cameron isolated in 2011 over a proposed EU fiscal pact, having initially said she would “find a way” to accommodate his demands.

She also offered to help Mr Cameron stop the appointment of Jean-Claude Juncker as European Commission president last year, only to leave the British leader virtually alone when she switched sides.

“Cameron doesn’t understand that for Merkel the EU is existential — it is a fact of history and geography — it is not optional,” she said. The Greek debt crisis and the conflict in Ukraine are reminders that Ms Merkel has other concerns.

Will Ms Merkel deliver for Mr Cameron on the biggest question of all? Observers of the last meeting between the two leaders noted how carefully the prime minister explored Ms Merkel’s willingness to back a UK deal.

Downing Street is carefully calibrating demands to ensure they do not stray across any Merkel red lines; indeed Mr Cameron’s big speech on EU migration last year was cleared in advance with Berlin.

But Mr Cameron has also been careful to follow the advice of Ms Merkel that he needs other allies; the British prime minister deliberately chose to visit Paris and Warsaw last month before touching down in Berlin.

Indeed by the start of the EU summit in Brussels he will have spoken to each of his fellow 27 leaders, along with Donald Tusk, the European Council president, and Mr Juncker, as well as leading MEPs.

Charles Grant, director of the Centre for European Reform, says: “Germans tell me that Merkel will lead from behind on this. She will work through Tusk and Juncker and keep the French by her side so people don’t have the perception she’s running the show.”